Lab for Automated Reasoning and Analysis LARA

Forward and backward propagation

Let denote the result of propagating backward the error conditions, that is, the negation of postcondition. We define by

for all . Here for any set of states we write for the complement of , that is, the set .

If is the postcondition operator, that is,

1. Then show that

where denotes the inverse of relation . In other words, computing weakest preconditions corresponds to propagating possible errors backwards.

Galois Connection

In the lecture we talked about a special case of Galois connection, called Galois insertion. Galois connection is in general defined by two monotonic functions and between partial orders on and on , such that

for all and (intuitively, the condition means that is approximated by ).

2. Show that the condition is equivalent to the conjunction of these two conditions:

hold for all and .

3. Let and satisfy the condition of Galois connection. Show that the following three conditions are equivalent: